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Can you include more information, like a link to the text in question? There are a couple possible meanings that could apply.
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Kristina LopezJul 7 '13 at 17:27

The context of the use of "danced his way out of poverty" is not the same meaning as Shaona's answer. In your linked story, Johnny Burgess, Jr. used his dancing skills to gain fame, giving him the opportunity to get out of poverty...thereby "dancing his way out of poverty".
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Kristina LopezJul 7 '13 at 17:55

3 Answers
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For the example you linked to, I think Kristina has explained the meaning in her comment.

Shoana suggests that dance, as a metaphor, simply means "succeeded", but I wouldn't expect a metaphor like this to have such a simple translation.

You originally referred to a different example - danced his way out of prison - which may not fit either of these interpretations.

Dance sometimes is referred to as "fancy footwork". Here is what the Free Dictionary Online says about fancy footwork. The third definition might fit you example:

(literally) clever and intricate dance steps. The old man was known for his fancy footwork when he was on Broadway.

(literally) adroit movements of the feet that help someone retain balance or move through treacherous territory. It took some fancy footwork to get down the mountain carrying the injured child.

(figuratively) a clever and intricate strategy that helps someone get out of trouble.The governor did some fancy footwork to keep from getting blamed for the scandal.

"Dancing" is used similarly as when someone is in trouble making a case or justifying their statements. They might have to do some "fast dancing" or "do a tap dance" (think fast and act fast) to bolster their position . This expression can also be taken negatively to mean to create a distraction and avoid scrutiny.

While Johnny Burgess might have become a talented dancer, and thereby escaped a life of poverty, I have a difficult time seeing how that talent (literally) would help someone get out of prison. I would guess that you were not aware of the clever strategy definition of dance, and didn't recognize that the Johnny Burgess story may not really fit your context. That being said, you might want to say more about that specific context, just so we can be sure.

"Walk" is another "escape" metaphor that you might be interested in. To say that a prisoner or criminal suspect "walks" can mean (Free Dictionary) that they are released from custody or acquitted of a crime. It's possible that "dance out of prison" is a variation of "walk out of prison", with the added notion of cleverness on the prisoner's part.

The apparent meaning may make it seem that Robert danced while going out of the prison, but it actually means that Robert succeeded in going out of the prison.
Coming out of a prison is akin to coming out of hell. So they may have put it in this way to express his gusto in doing so.

The X in question can be just about anything, the idiom means that you used X to get out of Y. For example:

He fought his way out of the alley.
She sang his way to the top of the charts.
He lied his way into office.
She can't act her way out of a paper bag.

There is also another, similar formulation:

To X yourself to Y

For example:

He drank himself to an early grave.
She ran herself to exhaustion.

So, to dance oneself out of prison would mean to use dance as a way of getting out of prison. Not necessarily by escaping, in fact, in the absence of context, I would read it to mean that his dancing showed him to be rehabilitated and helped him get released on parole.